Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model

Abstract

In this paper we investigate the optimal control approach for the active control and drag optimization of incompressible viscous flow past circular cylinders. The control function is the time angular velocity of the rotating cylinder. The wake flow is solved in the laminar regime (Re = 200) with a finite element method. Due to the CPU and memory costs related to the optimal control theory, a Proper Orthogonal Decomposition (POD) Reduced Order Model (ROM) is used as the state equation. The key enablers to an accurate and robust POD ROM are the introduction of a time dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem and the use of a snapshot ensemble for POD based on chirp-forced transients. Since the POD basis represents only velocities, we minimize a drag-related cost functional characteristic of the wake unsteadiness. The optimization problem is solved using Lagrange multipliers to enforce the constraints. 25% of relative drag reduction is found when the Navier-Stokes equations are controlled using an harmonic control function deduced from the optimal solution determined with the POD ROM. Earlier numerical studies concerning mean drag reduction are confirmed: it is shown in particular that without a sufficient penalization of the control input, our approach is energetically inefficient. The main result is that a cost reduction factor of one hundred and 760 is obtained for the CPU time and the memory respectively. Finally, limits of the performance of our approach are discussed.

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models

“…Following this approach the POD functions are determined once for all at the beginning of the optimization process and no refresh is realized. This method was successfully applied to control the cylinder wake flow in [15]. It was demonstrated that an accurate and robust POD ROM can be derived using a snapshot ensemble for POD based on chirp-forced transients of the flow.…”

“…Since the POD basis is intrinsic to a particular flow, we need to give special attention to adapt the POD ROM (and the POD basis naturally) to changes in physics when the flow is altered by control. This central question is discussed in more details in [15] where two strategies are evidenced for use of POD ROM in an optimization setting. A first approach consists in distributing uniformly in the control parameter space the snapshot ensemble to be used for POD.…”

“…1). In addition, as opposed to what was made in [15], where the cost functional to be minimized was not the drag but a drag-related cost function (the turbulent kinetic energy contained in the wake), we will directly take here for cost functional the mean drag coefficient (viscous and pressure contributions). Then, to reduce as much as possible the computational costs associated to the present study, the flow is considered two-dimensional and in the laminar regime.…”

“…Then, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations, the so-called global convergence 3 of the trust-region methods. Moreover, in [15], the POD basis used to derive the reduced-order models represented only velocities. Therefore, a drag-related cost functional characteristic of the wake unsteadiness was minimized.…”

On least-order flow representations for aerodynamics and aeroacoustics

“…On least-order flow representations for aerodynamics and aeroacoustics 371 force-related structures (Protas & Wesfreid 2003;Bergmann, Cordier & Brancher 2005), this constitutes an exercise of a first check of OID's physical plausibility. A major goal of the modelling efforts of this paper is to provide a physical understanding of shear flow noise generation.…”